JEE Main 2021 · 27 Jul 2021 · Shift 1

Q1.Two identical tennis balls each having mass 'm' and charge 'q' are suspended from a fixed point by threads of length 'ℓ'. What is the equilibrium separation when each thread makes a small angle 'θ' with the vertical?

Q2.Match the moment of inertia (MI) of rods with different configurations:

Choose the correct answer from the options given below:

Q3.The figure shows two solid discs with radius $R$ and $r$ respectively. If mass per unit area is same for both, what is the ratio of MI of bigger disc around axis AB (which is ⟂ to the plane of the disc and passing through its centre) to MI of smaller disc around one of its diamet …

Q4.In the given figure, a battery of emf E is connected across a conductor PQ of length 'ℓ' and different area of cross-sections by having radii $r_1$ and $r_2$ ($r_2<r_1$).

Choose the correct option as one moves from P to Q:

Q5.A capacitor of capacitance $\mathrm{C}=1\mu \mathrm{F}$ is suddenly connected to a battery of 100 volt through a resistance $\mathrm{R}=100\Omega$. The time taken for the capacitor to be charged to get 50 V is:
[Take $\ln 2=0.69$]

Q6.Three objects A, B and C are kept in a straight line on a frictionless horizontal surface. The masses of A, B and C are $m$, $2m$ and $2m$ respectively. A moves towards B with a speed of $9\mathrm{m}/\mathrm{s}$ and makes an elastic collision with it. Thereafter B makes a com …

Q7.In the reported figure, a capacitor is formed by placing a compound dielectric between the plates of parallel plate capacitor. The expression for the capacity of the said capacitor will be: (Given area of plate $=\mathrm{A}$)

Q8.A light cylindrical vessel is kept on a horizontal surface. Area of base is A. A hole of cross-sectional area 'a' is made just at its bottom side. The minimum coefficient of friction necessary to prevent sliding the vessel due to the impact force of the emerging liquid is …

Q9.The relative permittivity of distilled water is 81. The velocity of light in it will be: (Given ${\mu }_{\mathrm{r}}=1$)

Q10.Assertion A: If A, B, C, D are four points on a semicircular arc with centre at 'O' such that $|\overrightarrow{\mathrm{AB}}|=|\overrightarrow{\mathrm{BC}}|=|\overrightarrow{\mathrm{CD}}|$, then …

Q11.The number of molecules in one litre of an ideal gas at 300 K and 2 atmospheric pressure with mean kinetic energy $2\times 10^{-9}\mathrm{J}$ per molecule is:

Q12.In the reported figure, there is a cyclic process ABCDA on a sample of 1 mol of a diatomic gas. The temperature of the gas during the process $\mathrm{A}\to \mathrm{B}$ and $\mathrm{C}\to \mathrm{D}$ are ${\mathrm{T}}_1$ and $T_2$ ($T_1>T_2$) respectively. …

Q13.If '$f$' denotes the ratio of the number of nuclei decayed ($N_d$) to the number of nuclei at $t=0$ ($N_0$) then for a collection of radioactive nuclei, the rate of change of '$f$' with respect to time is given as: [$\lambda$ is the radioactive decay constant]

Q14.Assertion A: If in five complete rotations of the circular scale, the distance travelled on main scale of the screw gauge is 5 mm and there are 50 total divisions on circular scale, then least count is 0.001 cm.

Reason R: Least Count …

Q15.Two capacitors of capacities 2C and C are joined in parallel and charge up to potential V. The battery is removed and the capacitor of capacity C is filled completely with a medium of dielectric constant K. The potential difference across the capacitors will now be:

Q16.In Young's double slit experiment, if the source of light changes from orange to blue then:

Q17.A ball is thrown up with a certain velocity so that it reaches a height '$h$'. Find the ratio of the two different times of the ball reaching $\frac{\mathrm{h}}{3}$ in both the directions.

Q18.A particle starts executing simple harmonic motion (SHM) of amplitude 'a' and total energy E. At any instant, its kinetic energy is $\frac{3E}{4}$ then its displacement '$y$' is given by:

Q19.A 0.07 H inductor and a $12\Omega$ resistor are connected in series to a $220\mathrm{V}$, $50\mathrm{Hz}$ ac source. The approximate current in the circuit and the phase angle between current and source voltage are respectively. [Take $\pi$ as $\frac{22}{7}$]

Q20.A body takes 4 min. to cool from $61^{\circ }\mathrm{C}$ to $59^{\circ }\mathrm{C}$. If the temperature of the surroundings is $30^{\circ }\mathrm{C}$, the time taken by the body to cool from $51^{\circ }\mathrm{C}$ to $49^{\circ }\mathrm{C}$ is:

Q21.A transistor is connected in common emitter circuit configuration, then collector supply voltage is 10 V and the voltage drop across a resistor of $1000\Omega$ in the collector is 0.6 V. If the current gain factor $(\beta )$ is 24, then the base current is ______ $\mu \mathrm{A}$. …

Q22.A prism of refractive index ${\mathrm{n}}_1$ and another prism of refractive index ${\mathrm{n}}_2$ are stuck together (as shown in the figure). ${\mathrm{n}}_1$ and ${\mathrm{n}}_2$ depend on $\lambda$, the wavelength of light, according to the relation …

Q23.Consider an electrical circuit containing a two way switch '$S$'. Initially $S$ is open and then $T_1$ is connected to $T_2$. As the current in $\mathrm{R}=6\Omega$ attains a maximum value of steady state level, ${\mathrm{T}}_1$ is disconnected from ${\mathrm{T}}_2$ and immed …

Q24.A stone of mass 20 g is projected from a rubber catapult of length 0.1 m and area of cross section $10^{-6}{\mathrm{m}}^2$ stretched by an amount 0.04 m. The velocity of the projected stone is ______ $\mathrm{m}/\mathrm{s}$.
(Young's modulus of rubber …

Q25.A particle of mass $9.1\times 10^{-31}\mathrm{kg}$ travels in a medium with a speed of $10^6\mathrm{m}/\mathrm{s}$ and a photon of a radiation of linear momentum $10^{-27}\mathrm{kg}\mathrm{m}/\mathrm{s}$ travels in vacuum. The wavelength of photon is ______ times the wavelength of …

Q26.The amplitude of upper and lower side bands of A.M. wave where a carrier signal with frequency 11.21 MHz, peak voltage 15 V is amplitude modulate by a 7.7 kHz sine wave of 5 V amplitude are $\frac{a}{10}\mathrm{V}$ and $\frac{b}{10}\mathrm{V}$ respectively. Then the value of …

Q27.In a uniform magnetic field, the magnetic needle has a magnetic moment $9.85\times 10^{-2}\mathrm{A}/{\mathrm{m}}^2$ and moment of inertia $5\times 10^{-6}{\mathrm{kg}\mathrm{m}}^2$. If it performs 10 complete oscillations in 5 seconds then the magnitude of the magnetic field is ______ …

Q28.Suppose two planets (spherical in shape) of radii $R$ and $2R$, but mass M and 9M respectively have a centre to centre separation 8R as shown in the figure. A satellite of mass 'm' is projected from the surface of the planet of mass 'M' directly towards the centre of the second p …

Q29.In Bohr's atomic model, the electron is assumed to revolve in a circular orbit of radius $0.5\mathring{A}$. If the speed of electron is $2.2\times 10^6\mathrm{m}/\mathrm{s}$, then the current associated with the electron will be $\simeq$ ______ $\times 10^{-2}\mathrm{mA}$. [Take $\pi$ …

Q30.A radioactive sample has an average life of 30 ms and is decaying. A capacitor of capacitance $200\mu \mathrm{F}$ is first charged and later connected with resistor 'R'. If the ratio of charge on capacitor to the activity of radioactive sample is fixed with respect to time then …

Q1.Given below are two statements:
Statement I: Rutherford's gold foil experiment cannot explain the line spectrum of hydrogen atom.
Statement II: Bohr's model of hydrogen atom contradicts Heisenberg's uncertainty principle.
In the light of the above statements, choose the most appr …

Q2.For a reaction of order n, the unit of the rate constant is:

Q3.Which one among the following chemical tests is used to distinguish monosaccharide from disaccharide?

Q4.Staggered and eclipsed conformers of ethane are:

Q5.The number of geometrical isomers found in the metal complexes $\left[{\mathrm{PtCl}}_2{\left({\mathrm{NH}}_3\right)}_2\right]$, $\left[\mathrm{Ni}(\mathrm{CO}{)}_4\right]$, $\left[\mathrm{Ru}{\left({\mathrm{H}}_2\mathrm{O}\right)}_3{\mathrm{Cl}}_3\right]$ and …

Q6.Which one of the following compounds will give orange precipitate when treated with 2,4-Dinitrophenyl hydrazine?

Q7.Consider the above reaction and identify the product P:

Q8.Match List-I with List-II:

List-I (Drug):
(a) Furacin
(b) Arsphenamine
(c) Dimetone
(d) Valium

List-II (Class of Drug):
(i) Antibiotic
(ii) Tranquilizers
(iii) Antiseptic
(iv) Synthetic antihistamines

Choose the most appropriate match:

Q9.The parameters of the unit cell of substance are $a=2.5,b=3.0,c=4.0,\alpha =90^{\circ },\beta =120^{\circ },\gamma =90^{\circ }$. The crystal system of the substance is:

Q10.The oxidation states of 'P' in ${\mathrm{H}}_4{\mathrm{P}}_2{\mathrm{O}}_7$, ${\mathrm{H}}_4{\mathrm{P}}_2{\mathrm{O}}_5$ and ${\mathrm{H}}_4{\mathrm{P}}_2{\mathrm{O}}_6$, respectively are:

Q11.The type of hybridisation and magnetic property of the complex ${\left[{\mathrm{MnCl}}_6\right]}^{3-}$, respectively, are:

Q12.Presence of which reagent will affect the reversibility of the following reaction, and change it to a irreversible reaction:
${\mathrm{CH}}_4+{\mathrm{I}}_2\underset{\text{Reversible}}{\stackrel{h\nu }{\rightleftharpoons }}{\mathrm{CH}}_3\mathrm{I}+\mathrm{HI}$

Q13.The compound 'A' is a complementary base of ______ in DNA strands.

Q14.Match List-I with List-II:

List - I
(a) NaOH
(b) $\mathrm{Be}(\mathrm{OH}{)}_2$
(c) $\mathrm{Ca}(\mathrm{OH}{)}_2$
(d) $\mathrm{B}(\mathrm{OH}{)}_3$
(e) $\mathrm{Al}(\mathrm{OH}{)}_3$

List - II
(i) Acidic
(ii) Basic
(iii) Amphoteric

Choose the most appropriate answer f …

Q15.The correct order of stability of given carbocations is:

Q16.Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R.
Assertion R: Lithium halides are some what covalent is nature.
Reason R: Lithium possess high polarisation capability.
In the light of the above statements, choose the most appro …

Q17.Which one of the following statements is NOT correct?

Q18.The product obtained from the electrolytic oxidation of acidified sulphate solutions, is:

Q19.The statement that is INCORRECT about Ellingham diagram is:

Q20.Given below are two statements:
Statement I: Aniline is less basic than acetamide.
Statement II: In aniline, the lone pair of electrons on nitrogen atom is delocalised over benzene ring due to resonance and hence less available to a proton.
Choose the most appropriate option:

Q21.The conductivity of a weak acid HA of concentration $0.001\mathrm{mol}{\mathrm{L}}^{-1}$ is $2.0\times 10^{-5}\mathrm{S}{\mathrm{cm}}^{-1}$. If ${\wedge }_{\mathrm{m}}^{\circ }(\mathrm{HA})=190\mathrm{S}{\mathrm{cm}}^2{\mathrm{mol}}^{-1}$, the ionization constant ( …

Q22.The number of geometrical isomers possible in triamminetrinitrocobalt(III) is X and in trioxalatochromate(III) is Y. Then the value of $\mathrm{X}+\mathrm{Y}$ is ______.

Q23.In gaseous triethyl amine the "$-\mathrm{C}-\mathrm{N}-\mathrm{C}-$" bond angle is ______ degree.

Q24.An organic compound is subjected to chlorination to get compound A using 5.0 g of chlorine. When 0.5 g of compound A is reacted with ${\mathrm{AgNO}}_3$ [Carius Method], the percentage of chlorine in compound A is ______ when it forms 0.3849 g of AgCl
(Round off to the Nearest …

Q25.1.46 g of a biopolymer dissolved in a 100 mL water at 300 K exerted an osmotic pressure of $2.42\times 10^{-3}$ bar.
The molar mass of the biopolymer is ______ $\times 10^4$ g mol${}^{-1}$.
(Round off to the Nearest integer)
[Use : R $=0.083$ L bar mol${}^{-1}$ K${}^{-1}$]

Q26.$\mathrm{CO}{}_2$ gas adsorbs on charcoal following Freundlich adsorption isotherm. For a given amount of charcoal, the mass of $\mathrm{CO}{}_2$ adsorbed becomes 64 times when the pressure of $\mathrm{CO}{}_2$ is doubled. The value of $n$ in the Freundlich isotherm equation is ______ …

Q27.$\mathrm{PCl}{}_5\rightleftharpoons \mathrm{PCl}{}_3+\mathrm{Cl}{}_2$ $K_C=1.844$
3.0 moles of $\mathrm{PCl}{}_5$ is introduced in a 1 L closed reaction vessel at 380 K. The number of moles of $\mathrm{PCl}{}_5$ at equilibrium is ______ $\times 10^{-3}$. (Round off to the Nearest integer)

Q28.The difference between bond orders of CO and $\mathrm{NO}{}^{+}$ is $\frac{X}{2}$ where X = ______
(Round off to the Nearest integer)

Q29.The density of NaOH solution is 1.2 gm cm${}^{-3}$. The molality of this solution is ______ m.
(Round off to the Nearest integer)
[Use : Atomic masses : Na 23.0 u O: 16.0 u H: 1.0 u Density of $\mathrm{H}{}_2\mathrm{O}$: 1.0 g cm${}^{-3}$]

Q30.For water at $100^{\circ }$C and 1 bar
${\Delta }_{\text{vap}}H-{\Delta }_{\text{vap}}U=$ ______ $\times 10^2$ J mol${}^{-1}$
(Round off to the Nearest integer)
[Use: R = 8.31 J mol${}^{-1}$ K${}^{-1}$]
[Assume volume of $\mathrm{H}{}_2\mathrm{O}$(l) is much smaller than volume of $\mathrm{H}{}_2\mathrm{O}$(g). Assume …

Q1.Let C be the set of all complex numbers. Let
$S_1=\left\{z\in C\Vert |z-3-2i{|}^2=8\right\}$,
$S_2=\{z\in C\mid \mathrm{Re}(z)\ge 5\}$ and
$S_3=\{z\in C\mid |z-\bar{z}|\ge 8\}$.
Then the number of element in …

Q2.If $\sin \theta +\cos \theta =\frac{1}{2}$, then $16(\sin (2\theta )+\cos (4\theta )+\sin (6\theta ))$ is equal to :

Q3.Let $\vec{a}=\hat{i}+\hat{j}+2\hat{k}$ and $\vec{b}=-\hat{i}+2\hat{j}+3\hat{k}$. Then the vector product $(\vec{a}+\vec{b})\times ((\vec{a}\times ((\vec{a}-\vec{b})\times \vec{b}))\times \vec{b})$ is equal to :

Q4.Let the plane passing through the point $(-1,0,-2)$ and perpendicular to each of the planes $2x+y-z=2$ and $x-y-z=3$ be $ax+by+cz+8=0$. Then the value of $a+b+c$ is equal to :

Q5.Let $f:\left(-\frac{\pi }{4},\frac{\pi }{4}\right)\to R$ be defined as

If $f$ is continuo …

Q6.The compound statement $(P\vee Q)\wedge (\sim P)\Rightarrow Q$ is equivalent to :

Q7.Let
$A=\left\{(x,y)\in R\times R\mid 2x^2+2y^2-2x-2y=1\right\}$,
$B=\left\{(x,y)\in R\times R\mid 4x^2+4y^2-16y+7=0\right\}$ and
$C=\left\{(x,y)\in R\times R\mid x^2+y^2-4x-2y+5\le r^2\right\}$.
Then the minimum value of $|r|$ such that …

Q8.Let $A=\left[\begin{array}{cc}1& 2\\ -1& 4\end{array}\right]$. If $A^{-1}=\alpha I+\beta A$, $\alpha ,\beta \in R$, $I$ is a $2\times 2$ identity matrix, then $4(\alpha -\beta )$ is equal to :

Q9.Let $y=y(x)$ be solution of the differential equation ${\log }_e\left(\frac{dy}{dx}\right)=3x+4y$, with $y(0)=0$. If $y\left(-\frac{2}{3}{\log }_{e}2\right)=\alpha {\log }_{e}2$, then the value of $\alpha$ is equal to :

Q10.A ray of light through $(2,1)$ is reflected at a point $P$ on the $y$-axis and then passes through the point $(5,3)$. If this reflected ray is the directrix of any ellipse with eccentricity $\frac{1}{3}$ and the distance of the nearer focus from this directrix is …

Q11.The value of ${\lim }_{n\to \infty }\frac{1}{n}{\sum }_{j=1}^n\frac{(2j-1)+8n}{(2j-1)+4n}$ is equal to :

Q12.If the area of the bounded region $R=\left\{(x,y):\max \left\{0,{\log }_{e}x\right\}\le y\le 2^x,\frac{1}{2}\le x\le 2\right\}$ is $\alpha {\left({\log }_{e}2\right)}^{-1}+\beta \left({\log }_{e}2\right)+\gamma$, then the value of $(\alpha +\beta -2\gamma {)}^2$ is equal to …

Q13.If the coefficients of $x^7$ in ${\left(x^2+\frac{1}{bx}\right)}^{11}$ and $x^{-7}$ in ${\left(x-\frac{1}{b{x}^2}\right)}^{11},b\ne 0$ are equal, then the value of $b$ is equal to :

Q14.Two tangents are drawn from the point $P(-1,1)$ to the circle $x^2+y^2-2x-6y+6=0$. If these tangents touch the circle at points A and B , and if D is a point on the circle such that length of the segments $AB$ and $AD$ are equal, then the area of the triangle …

Q15.The probability that a randomly selected 2-digit number belongs to the set $\{n\in N:\left(2^n-2\right)$ is a multiple of 3$\}$ is equal to :

Q16.Let P and Q be two distinct points on a circle which has center at $\mathrm{C}(2,3)$ and which passes through origin O . If OC is perpendicular to both the line segments CP and CQ , then the set $\{\mathrm{P},\mathrm{Q}\}$ is equal to :

Q17.The value of the definite integral ${\int }_{-\frac{\pi }{4}}^{\frac{\pi }{4}}\frac{dx}{\left(1+e^{x\cos x}\right)\left({\sin }^{4}x+{\cos }^{4}x\right)}$ is equal to :

Q18.Let $f:R\to R$ be a function such that $f(2)=4$ and $f^{\prime }(2)=1$. Then the value of ${\lim }_{x\to 2}\frac{x^{2}f(2)-4f(x)}{x-2}$ is equal to :

Q19.Let $\alpha ,\beta$ be two roots of the equation $x^2+(20{)}^{\frac{1}{4}}x+(5{)}^{\frac{1}{2}}=0$ Then ${\alpha }^8+{\beta }^8$ is equal to :

Q20.If the mean and variance of the following data :
$6,10,7,13,a,12,b,12$ are 9 and $\frac{37}{4}$ respectively, then $(a-b{)}^2$ is equal to :

Q21.If $y=y(x),y\in \left[0,\frac{\pi }{2}\right]$ is the solution of the differential equation $\sec y\frac{dy}{dx}-\sin (x+y)-\sin (x-y)=0$, with $y(0)=0$, then $5y^{\prime }\left(\frac{\pi }{2}\right)$ is equal to ____ .

Q22.For real numbers $\alpha$ and $\beta$, consider the following system of linear equations:
$x+y-z=2,x+2y+\alpha z=1,2x-y+z=\beta$.
If the system has infinite solutions, then $\alpha +\beta$ is equal to ____ .

Q23.Let a plane P pass through the point $(3,7,-7)$ and contain the line, $\frac{\mathrm{x}-2}{-3}=\frac{\mathrm{y}-3}{2}=\frac{\mathrm{z}+2}{1}$. If distance of the plane $P$ from the origin is $d$, then $d^2$ is equal to ____ .

Q24.Let $F:[3,5]\to R$ be a twice differentiable function on ( 3,5 ) such that $F(x)=e^{-x}{\int }_3^x\left(3t^2+2t+4F^{\prime }(t)\right)dt$.
If $F^{\prime }(4)=\frac{\alpha {\mathrm{e}}^{\beta }-224}{{\left({\mathrm{e}}^{\beta }-4\right)}^2}$, then $\alpha +\beta$ …

Q25.Let $f(x)=\left|\begin{array}{ccc}{\sin }^{2}x& -2+{\cos }^{2}x& \cos 2x\\ 2+{\sin }^{2}x& {\cos }^{2}x& \cos 2x\\ {\sin }^{2}x& {\cos }^{2}x& 1+\cos 2x\end{array}\right|,x\in [0,\pi ]$. Then the maximum value of $f(x)$ is equal to ____.

Q26.Let $\vec{a}=\hat{i}+\hat{j}+\hat{k},\vec{b}$ and $\vec{c}=\hat{j}-\hat{k}$ be three vectors such that $\vec{a}\times \vec{b}=\vec{c}$ and $\vec{a}\cdot \vec{b}=1$. If the length of projection vector of the vector $\vec{b}$ on the vector $\vec{a}\times \vec{c}$ is $\ell$, the …

Q27.Let $f:[0,3]\to R$ be defined by $f(x)=\min \{x-[x],1+[x]-x\}$
Where $[x]$ is the greatest integer less than or equal to $x$.
Let $P$ denote the set containing all $x\in [0,3]$ where $f$ is discontinuous, and $Q$ denote the set containing all $x\in (0,3)$ where $f$ is n …

Q28.Let $S=\{1,2,3,4,5,6,7\}$. Then the number of possible function $f:S\to S$ such that $f(m\cdot n)=f(m)\cdot f(n)$ for every $m,n\in S$ and $m\cdot n\in S$ is equal to ____.

Q29.Let the domain of the function
$f(x)={\log }_4\left({\log }_5\left({\log }_3\left(18x-x^2-77\right)\right)\right)$ be $(a,b)$.
Then the value of the integral ${\int }_a^b\frac{{\sin }^{3}x}{\left({\sin }^{3}x+{\sin }^3(a+b-x)\right)}dx$ is equal to ____.

Q30.If ${\log }_{3}2,{\log }_3\left(2^x-5\right),{\log }_3\left(2^x-\frac{7}{2}\right)$ are in an arithmetic progression, then the value of $x$ is equal to ____.